Comparing many distributions

The extended movies dataset

import altair as alt
from vega_datasets import data

movies_extended = data.movies().dropna(subset=['Major_Genre'])
movies_extended

	Title	US_Gross	Worldwide_Gross	US_DVD_Sales	...	Director	Rotten_Tomatoes_Rating	IMDB_Rating	IMDB_Votes
1	First Love, Last Rites	10876.0	10876.0	NaN	...	None	NaN	6.9	207.0
2	I Married a Strange Person	203134.0	203134.0	NaN	...	None	NaN	6.8	865.0
3	Let's Talk About Sex	373615.0	373615.0	NaN	...	None	13.0	NaN	NaN
...	...	...	...	...	...	...	...	...	...
3198	Zoom	11989328.0	12506188.0	6679409.0	...	Peter Hewitt	3.0	3.4	7424.0
3199	The Legend of Zorro	45575336.0	141475336.0	NaN	...	Martin Campbell	26.0	5.7	21161.0
3200	The Mask of Zorro	93828745.0	233700000.0	NaN	...	Martin Campbell	82.0	6.7	4789.0

2926 rows × 16 columns

For this exercise, we will use an extended movies dataset containing additional genres.

We load in that dataset from the vega sample repository and drop all the rows that have a NaN value in the column "Major Genre", since we are only interested in comparing the movies that we know belong to a genre.

The question we will try to answer in this slide deck is “Which genres have the highest worldwide gross?”.

Many distributions can’t be effectively compared with histograms

alt.Chart(movies_extended).mark_bar().encode(
    alt.X('Worldwide_Gross', bin=alt.Bin(maxbins=30)),
    alt.Y('count()'),
    alt.Color('Major_Genre'))

As we have seen, histograms are not very effective for comparing multiple distributions, and not at all with this many different groups.

Most genres have a low worldwide gross and because our histograms are stacked on top of each other the left-most bar dominates with around 2000 movies. It extends the y-axis so that it is hard to see how many observations there are for the highest x-values (but we know there are some since the axis extends that far).

Many distributions can’t be effectively compared with densities either

(alt.Chart(movies_extended).mark_area().transform_density(
    'Worldwide_Gross',
    groupby=['Major_Genre'],
    as_=['Worldwide_Gross', 'density'])
 .encode(
    alt.X('Worldwide_Gross'),
    alt.Y('density:Q'),
    alt.Color('Major_Genre')))

Although we saw previously that a layered density chart is better than a stacked histogram, it is not effective in comparing this many distributions either. Particularly not when the bulk of the distribution is concentrated in a small area, such as the low x-values in this plot.

This density chart shares many of the issues with the histogram from the previous slide, and this visualization is a poor choice for our data.

We could try faceting vertically with one density or histogram per facets, but there would be a lot of subplots with this many genres and they might not be that easy to compare when they are in facets far apart.

So how can we create an effective comparison between all the genres?

Bar charts are effective for comparing a single value per group but hides variation

alt.Chart(movies_extended).mark_bar().encode(
    alt.X('mean(Worldwide_Gross)'),
    alt.Y('Major_Genre'))

We could use a barplot if we were only interested in comparing a single value from each distribution, such as the mean or median.

However, this is generally not a good idea because any plot that only shows a single value from a distribution will hide the variation in the data, which could lead us to arrive at incorrect conclusions as you will see in the next slide.

Showing a single value can lead to incorrect conclusions

Beyond Bar and Line Graphs: Time for a New Data Presentation Paradigm

This image illustrates how the same bar chart (A) could have been generated from three different distributions (B, C, or D).

If we just looked at the bar chart, we could not tell which of the distributions it came from, but looking at the points directly we would arrive at a widely different conclusion regarding our data.

In sample B, the points seem to on average be slightly different between the two groups, although there is significant overlap between the distributions.

In sample C there seems to be one outlier data point that increases the mean significantly for its group, but most of the other points are pretty similar between the two groups.

In D, both groups have their data distributed bimodally (in two peaks). Maybe this means that there is a third group in this data to which all the high values belong too?

All this valuable information about our data, is lost when visualizing it as a bar chart. Remember, bar charts are best suited for displaying individual values such as counts, proportions, and sums.

To arrive at more accurate conclusions we need to present a richer representation of the data such as a histogram or a density plot, which are great as long as we don’t have too many distributions to compare.

But which visualizations could we use to accurately represent the distributions while still remaining effective for comparing many distributions in the same plot?

Showing individual observations gives a richer representation than bar charts

alt.Chart(movies_extended).mark_tick().encode(
    alt.X('Worldwide_Gross'),
    alt.Y('Major_Genre'))

One approach would be to show the individual observations, such as in this chart. This is a compact plot, which makes it easy to compare the different genres, since they are close to each other in the chart.

We can also clearly see the reason why the x-axis goes up to such a large max value: There are two movies that grossed highly above the rest!

Which are these movies?

Tooltips are helpful for answering questions about specific observations

alt.Chart(movies_extended).mark_tick().encode(
    alt.X('Worldwide_Gross'),
    alt.Y('Major_Genre'),
    alt.Tooltip('Title:N'))

Altair really shines in answering questions like this! In addition to having a robust visualization grammar, it also provides a grammar for interactive features.

We will dive deep into more complex interactivity in the last module in this course, but for now, we can simply add a tooltip and answer the question we just posed by hovering with the mouse over the two highest grossing movies. Try it!

(We need the :N because there are some title values that makes Altair not recognize this dataframe column as nominal otherwise)

Although this visualization is useful in getting information about the individual movies, it is hard to tell exactly how many data points there are in the areas that are completely blue. Is there just enough so that we don’t see any of the white background or are there thousands of observations stacked on top of each other?

This plot is saturated, something we will discuss in detail in the next module. We could use transparency to alleviate this issue to some extent, but it is more effective to use colour for representing the counts, such as in a heatmap.

Heatmaps can compare multiple distributions without saturation

(alt.Chart(movies_extended).mark_rect().encode(
    alt.X('Worldwide_Gross', bin=alt.Bin(maxbins=100)),
    alt.Y('Major_Genre'),
    alt.Color('count()')))

A heatmap of the number of observations is very similar to a histogram, but the count is mapped to the colour instead of to the height on the y-axis.

This heatmap shows the histogram for each of the genres right next to each other so that they are easy to compare and contrast, by looking at the varying colours.

We can see that some genres appear to have most of their observations close to zero, such as Comedy and Drama, whereas others are more spread out, such as Adventure movies.

This is a pretty effective visualization, but since the colour scale is the same for each genre and some genres have many more observations than others, it can be difficult to accurately interpret the distribution of those genres that have few observations.

For example, we wouldn’t be able to distinguish between regions with 10-20 and 40-50 observations since the colours are so similar.

Remember that comparing positions is more effective than comparing colours, so how can we use position to compare multiple distributions?

Boxplots show several key statistics from a distribution

Jhguch at en.wikipedia via Wikimedia Commons

A boxplot is a mix between showing individual values and a few key summary statistics.

Instead of showing just the mean or median as with a bar plot, a box plot shows 5 summary statistics.

The box represents three values, the median in the middle and the lower and upper quartile at the edges This means that 50% of the data points lie within the box.

The lines extending from the box are called whiskers and they can represent a few different statistics. Sometimes, they are showing the min and the max (the range) of our data.

However, it is more common that they show the furthest points that are still within 1.5 x the “interquartile” range from the edges of the box. The interquartile range is the distance between the edges of the box.

Conventionally, any observations that fall outside the whiskers, are drawn out as individual points and are sometimes referred to as “outliers”, which are occasionally discarded.

However, what really is an outlier in your data and what you should do with them depends on the question you are asking. Maybe these data points are the most important in your dataset or they could be measurement errors. You should always look further and think carefully before discarding data as outliers.

How does it look when we use boxplots to answer our question?

Boxplots can effectively compare multiple distributions

bar = alt.Chart(movies_extended).mark_bar().encode(
    alt.X('mean(Worldwide_Gross)'),
    alt.Y('Major_Genre'))

box = alt.Chart(movies_extended).mark_boxplot().encode(
    alt.X('Worldwide_Gross'),
    alt.Y('Major_Genre'))

box | bar

Here we show a boxplot next to the barplot we created initially. The | operator works similarly to the + operator, but instead of layering charts on top of each other, it puts them next to each other.

We can see that the box plots provide us with the same clear positional comparison as in the barplot, but we now also have access to additional information about each genres’ distribution. We can ensure ourselves that the distributions are roughly the same shape and view potential outliers.

We can directly answer the question we posed at the beginning: the Adventure genre generates the highest revenue both in terms of the medians and the overall position of the boxes.

It seems like Action is the second most highly grossing genre, but after that, it becomes hard to compare. We could make this visualization even more effective by sorting the boxes according to their median value.

Sorted boxplots more effective for comparing similar distributions

genre_order = movies_extended.groupby(
    'Major_Genre')['Worldwide_Gross'].median().sort_values().index.tolist()
alt.Chart(movies_extended).mark_boxplot().encode(
    alt.X('Worldwide_Gross'),
    alt.Y('Major_Genre', sort=genre_order))

Altair does not support sorting boxplots via the “x” or “-x” shortcut.

Instead, we need to create a list of the genres, ordered by value.

We can do this with pandas by grouping the dataframe, computing the median for the grossing columns, sorting the values, and then extracting the index as a list.

Now the visualization is very effective! It is easy to compare genres with similar grossing since they are right next to each other in the plot.

If we wanted to look closer at the differences between the bulk of the distributions we could exclude the two most highly grossing movies.

Zooming in facilitates comparison of small differences

filtered_movies = movies_extended[movies_extended['Worldwide_Gross'] < 1_500_000_000]
alt.Chart(filtered_movies).mark_boxplot().encode(
    alt.X('Worldwide_Gross'),
    alt.Y('Major_Genre', sort=genre_order))

Here we can compare the distributions more clearly, and it really looks like there are no large differences in the medians between the five genres behind Adventure and Action.

We will need to keep in mind that there are two additional data points that we have excluded from this visualization, but neither the median line nor the box would be much affected by excluding just a few values from a large population.

If we would have shown the mean instead of the median, this could have been an issue since the mean can be affected significantly from just a few extreme values.

Boxplots can be scaled by the number of observations

alt.Chart(movies_extended).mark_boxplot().encode(
    alt.X('Worldwide_Gross'),
    alt.Y('Major_Genre', sort=genre_order),
    alt.Size('count()'))

The size of the boxes can be made proportional to the count of observations in that group.

This helps us see if there are many more observations underlying some of the boxes versus others. In our plot, we can see that the “comedy” genre has the most movies whereas there are few documentaries and black comedies.

However, it also makes it hard to tell where the lines of the median and the boxes are for many of the distributions. A better alternative could be to not scale the boxes and instead, include a bar chart with the number of observations per genre together with the boxplot.

Boxplots are not able to accurately represent data with multiple peaks

From Autodesk research

While boxplots are effective for visualizing multiple distributions, they also have their shortcomings.

One of their main downsides is that they are not effective in showing distributions with multiple peaks. This can be seen in the animation in this slide, where variation in the raw data does no result in any change in the boxplots.

Two effective visualizations for many distributions that also handle multiple peaks are violin plots (as shown in this slide) and stripplots.

Violinplots are similar to density plots put next to each other, and stripplots prevent overlap by distributing the points in a cloud instead of a straight line.

There are also sina plots, which combine the best of violin plots and strip plots together.

These are all highly effective, but not yet available to make easily in Altair, so we will not be teaching them here.

Boxplots are still very effective in comparing multiple distributions, and when there are even more distributions than what we have here, e.g. 50+, then boxplots are often easier to interpret than the alternatives mentions above, since they are simpler.

Let’s apply what we learned!